PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. Does $\hat{C}(E) = \mathcal{B}(E)$ imply $E$ is reflexive?
(This would in turn imply that $E$ is separable).

Some definitions:

A Banach space is a Grothendieck space if a sequence in $E'$ which is $\sigma(E', E)$-convergent is automatically $\sigma(E', E'')$-convergent.
Or equivalently: every $\sigma(E', E)$ zero sequence has subsequence which is $\sigma(E', E'')$-convergent,
or equivalently: every linear, bounded operator from $E$ to $c_0$ (or any separable Banach space) is automatically weakly compact.

The inclusion $\hat{C}(E) \subset \mathcal{B}(E)$ is trivially true. If $E$ is separable then $\hat{C}(E) = \mathcal{B}(E)$. [To see this use the Hahn-Banach theorem to show that $\mathcal{B}_E \in \hat{C}(E)$. As translations and scalar multplications are measurable with regard to the cylindrical $\sigma$-algebra the other inclusion follows.]

A reflexive space is automatically Grothendieck.

For a separable Grothendieck space $E$ we have that the identity is weakly compact so $E$ becomes reflexive

A reflexive space $E$ with $\hat{C}(E) = \mathcal{B}(E)$ is automatically separable ([1], Prop. 2.6, p.19).
Without reflexivity the equality $\hat{C}(E) = \mathcal{B}(E)$ does not imply $E$ is separable in general.

The example $E = \ell^2(\mathbb{R})$ shows that there is a reflexive and non-separable space with $\mathcal{B}(E) \not= \hat{C}(E)$

There are non-reflexive Grothendieck spaces which do not contain $\ell^{\infty}$ (cf. [3]). So we can't simply reduce to this case.
I don't know much more about Grothendieck spaces though or characterizations of them that might be helpful.

1 Answer
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Space $C(K)$ of continuous functions on a Stone space $K$ is Grothendieck, right? So take $K$ so large that countably many continuous functions do not separate points in $K$. Then (as in the $l^2(I)$ answer to Question 24432 cited) the weak Baire sets (= the cylindrical sigma-algebra) is not equal to the weak Borel sets, and certainly not equal to the norm Borel sets. Since the closed unit ball is not a weak Baire set.

I'm not sure I understand you correctly, but I wanted an example where the cylindrical algebra is equal to the Borel sigma algebra. In the $\ell^2(I)$ case and the $C(K)$ case for e.g. $K = \beta \mathbb{N}$ it is not, as we know from the other question. Is there some characterisation of Grothendieck spaces I'm not aware of you are using here? As in all non-reflexive Grothendieck spaces are of $C(K)$ type (where $K$ is Stonean)?
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santker hebolnJul 22 '10 at 6:08